Indirect Effects of Zebra Mussels (Dreissena Polymorpha) on the Planktonic Food Web

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Indirect effects of zebra mussels (Dreissena polymorpha) on the planktonic food web

Erik G. Noonburg, Brian J. Shuter, and Peter A. Abrams

Abstract: The exotic zebra mussel (Dreissena polymorpha) has caused dramatic reductions in phytoplankton density in lakes with dense mussel populations. However, the indirect effects of this invader on other trophic groups have been inconsistent and difficult to interpret. In some lakes, zebra mussels appear to have had little effect on zooplankton density, despite decreasing the abundance of their phytoplankton prey. We analyze food web models to test hypothesized mechanisms for the absence of a strong effect of dreissenids on zooplankton. Our results suggest that neither reduced inedible algal interference with zooplankton filtering nor reduced phytoplankton self-shading is sufficient to explain the insensitivity of zooplankton populations to dreissenid competition. Instead, we show how the impact of benthic filter feeders can be influenced by the rate of mixing within a basin, which limits phytoplankton delivery to the benthos. We explore the predictions of a simple spatially structured model and demonstrate that differences in abiotic factors that control mixing can result in large differences in direct and indirect effects of zebra mussel filtering. Résumé : Les lacs qui possèdent de grandes densités de moules zébrées (Dreissena polymorpha), une espèce exotique, subissent de fortes réductions de la densité du phytoplancton. Cependant, les effets indirects de cette espèce envahis- sante sur les autres groupes trophiques sont variables et difficiles à interpréter. Dans certains lacs, les moules zébrées semblent avoir peu d’effet sur la densité du zooplancton, malgré le déclin de l’abondance du phytoplancton qui leur sert de nourriture. Des modèles de réseaux alimentaires nous permettent de vérifier les mécanismes suggérés pour expliquer l’absence d’effets significatifs des dreissénidés sur le zooplancton. Nos résultats montrent que ni la réduction de l’interférence des algues non comestibles sur la filtration du zooplancton, ni la diminution de l’auto-ombrage par le phytoplancton ne sont assez importantes pour expliquer que les populations de zooplancton ne soient pas affectées par la compétition des dreissenidés. Plutôt, l’impact des filtreurs benthiques est influencé par le degré de brassage dans le bassin, ce qui limite l’apport de phytoplancton au benthos. L’examen des prédictions d’un modèle simple à structure spatiale nous permet de démontrer que des différences dans les facteurs abiotiques qui contrôlent le brassage peuvent amener des différences importantes dans les effets directs et indirects de la filtration des moules zébrées. [Traduit par la Rédaction] Noonburg et al. 1368

Introduction effects and multiple hypothesized indirect effects (Nalepa and Schloesser 1993; Strayer et al. 1999). Zebra mussels Biological invasions pose a serious threat to native bio- (Dreissena polymorpha) were first reported in Lake St. Clair diversity as well as humans’ ability to manage ecosystems in 1989 (Hebert et al. 1989) and have since established pop- for recreation and resource exploitation, yet managers are ulations in several drainages of eastern North America often unable to prevent the arrival of new species (Czech and (Johnson and Carlton 1996). In shallow bays and lake basins Krausman 1997; Mack et al. 2000; Pimentel et al. 2000). with dense mussel populations, estimates of total filtering Understanding and predicting the effects of exotic species rate are extremely high. For instance, Bailey et al. (1999) es- are therefore crucial research goals (Parker et al. 1999). The timated the time to filter the entire water column in part of direct effects of exotic species are often obvious, e.g., the re- the Bay of Quinte, Lake Ontario, to be as short as 0.05 days. placement of competitors or the extirpation of prey. How- Not surprisingly, in areas with sufficient suitable benthic ever, the indirect effects of exotics can be unexpected and habitat to support dense mussel populations, dreissenid fil- the causal links may be difficult to demonstrate. tering is generally associated with declines in phytoplankton The introduction of dreissenid mussels to North America abundance (e.g., Leach 1993; Nalepa et al. 1999; Idrisi et al. is an excellent example of an invasion with dramatic direct 2001). In contrast, despite potentially strong competition be-

Received 8 October 2002. Accepted 31 August 2003. Published on the NRC Research Press Web site at http://cjfas.nrc.ca on 21 November 2003. J17135 E.G. Noonburg1 and P.A. Abrams. Department of Zoology, University of Toronto, 25 Harbord Street, Toronto, ON M5S 3G5, Canada. B.J. Shuter. Department of Zoology, University of Toronto, 25 Harbord Street, Toronto, ON M5S 3G5, Canada, and Aquatic Research and Development, Ontario Ministry of Natural Resources, Lake Erie Management Unit, Box 429, Port Dover, ON N0A 1N0, Canada.

1Corresponding author (e-mail: [email protected]).

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1354 Can. J. Fish. Aquat. Sci. Vol. 60, 2003

tween dreissenids and zooplankton for phytoplankton, im- of water column mixing on the food web response to pacts on zooplankton vary from site to site. In some lakes, dreissenid filtering. We show that benthic–pelagic structure sampling data show no significant change in zooplankton can explain why dreissenid invasions do not produce a strong, density after dreissenids invade, despite decreases in phyto- consistent response in the zooplankton community. We pre- plankton density over the same time period (Wu and Culver dict that the responses of phytoplankton and zooplankton to 1991; Idrisi et al. 2001). Indeed, Wu and Culver (1991) sug- mussel filtering are sensitive to physical factors that deter- gested that zooplankton continued to control phytoplankton mine the water column mixing rate. dynamics in the western basin of Lake Erie even after the establishment of dense zebra mussel beds. This interpreta- tion has been challenged (MacIsaac et al. 1992; Nicholls and Model 1: a single, well-mixed compartment Hopkins 1993), and other investigators have suggested that zooplankton declined in response to a dreissenid invasion in Structure some systems (e.g., Bridgeman et al. 1995; Johannsson et al. We model the dynamics of phytoplankton, zooplankton, 2000). and a limiting nutrient in a small lake basin or in a bay that Two general types of mechanisms might explain the ab- is homogeneous but distinct from a larger lake. Water enters sence of a consistent negative effect of zebra mussel inva- the system from rivers or runoff (inflow contains nutrients, sions on zooplankton abundance. The first type involves but no organisms) and leaves at the same rate, resulting in indirect positive effects on phytoplankton productivity that constant volume. We assume that the limiting nutrient in the offset the increased consumption of phytoplankton by system is phosphorus and measure the abundance of all dreissenids. For instance, Padilla et al. (1996) constructed a trophic groups in units of phosphorus concentration. food web model and predicted that zebra mussel filtering Throughout our analysis, we assume that dreissenid bio- would disproportionately lower the density of phytoplankton mass is not dynamically linked to phytoplankton density. In- taxa that are inedible to zooplankton, thereby relaxing nutri- stead, we treat mussel abundance as an independent variable ent competition and increasing zooplankton foraging effi- and calculate its effect on the equilibrium concentrations of ciency, thus permitting a less abundant but more productive the other food web components. Our rationale is that dreis- and edible phytoplankton community to support equal or senids are limited by suitable hard benthic substrate. Dreis- even higher abundances of zooplankton. A second example senid density has been observed to fluctuate within these of this sort of mechanism is provided by Idrisi et al. (2001), habitats (Nalepa et al. 1995), and dreissenids have colonized who suggested that the increases in water clarity that accom- some areas with soft sediment (Dermott and Munawar 1993; pany mussel colonization could stimulate phytoplankton pro- Coakley et al. 1997; Berkman et al. 1998); however, differ- ductivity, thus maintaining existing zooplankton abundances. ences in suitable habitat appear to be the primary determi- nant of differences in mussel abundance among lake regions The second class of mechanism involves physical factors (Mellina and Rasmussen 1994). that potentially constrain the direct effects of mussel filtering on phytoplankton populations. In particular, zebra mussels We divide phytoplankton into two classes based on whether attached to the benthos are isolated to some degree from they are too large to be ingested by cladocerans. Characteris- their pelagic prey. Studies of vertical chlorophyll profiles tics of the two phytoplankton groups generate a trade-off be- and mussel ingestion rates in the field suggest that mussels tween vulnerability to zooplankton filtering and competitive refilter the lower portion of the water column more rapidly ability: small phytoplankton, which are edible to zooplank- than phytoplankton are delivered by mixing, and this might ton, are superior to large phytoplankton in nutrient competi- play a critical role in determining the impact of mussels on tion (reviewed in Andersen 1997). phytoplankton (MacIsaac et al. 1999; Yu and Culver 1999; We provide a brief description of the key model compo- Ackerman et al. 2001). Furthermore, dense dreissenid popu- nents here; the full equations appear in Table 1 and the pa- lations typically occur on hard substrates, and these can be rameters are defined in Table 2. The model describes the patchily distributed across the bottom of a lake. Hence, mus- dynamics of available phosphorus (R), large (L) and small sel consumption of phytoplankton may be limited by vertical (S) phytoplankton, and zooplankton (Z) in terms of the con- and horizontal mixing of the water column. centration of phosphorus associated with each of these components. The rate of change in R is the sum of contributions We evaluate these hypotheses using two food web models. from abiotic processes (inflow and outflow) and biotic pro- First, we derive predictions for the effects of dreissenid inva- cesses (plankton growth and phosphorus recycling): sions on the equilibrium abundances of zooplankton, phytoplankton, and phosphorus in a simple, well-mixed model of (1) dR/dt = I + E – G – O the pelagic food web. We then modify the model to incorpo- R R R R

rate inedible algal interference and phytoplankton self-shading where IR is inflow, OR is outflow, ER is the sum of excretion and determine the effects of these mechanisms by compari- and other losses from all organisms, and GR is phytoplankton son with results from the basic model. Using the western ba- population growth. Dynamics of small and large phyto- sin of Lake Erie as a case study, we show that primary plankton are determined by the difference between popula- production is insufficient to support the planktonic food web tion growth (GS, GL) and losses to filtering (FS, FL), outflow in the presence of a dense population of zebra mussels filter- (OS, OL), and sinking and other mortality (ES, EL): ing a well-mixed water column. Next, we address the hy- pothesis that isolation of mussels from their pelagic prey (2) dS/dt = GS – FS – ES – OS limits the effect of dreissenids on zooplankton. We introduce simple spatial structure to the model and explore the effects (3) dL/dt = GL – FL – EL – OL

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Table 1. Dynamic equations for the four components of the food chain model: available phosphorus (R), edible phytoplankton (S), inedible phytoplankton (L), and zooplankton (Z). Available phosphorus, R dR = I + E – G – O dt R R R R Inflow IR = QRin ()1 − ε aSZ Total plankton losses E = l S + l L + l Z + S + fbM(S + L) R S L Z ++ 1 aTSSS aTL LL

Total phytoplankton growth GR = GS + GL

Outflow OR =QR Edible phytoplankton, S dS = G – F – E – O dt S S S S VSR Growth G = S S + kRS

aSZ Filtering loss F = S + bMS S ++ 1 aTSSS aTL LL

Other losses ES =lSS

Outflow OS =QS Inedible phytoplankton, L dL = G – F – E – O dt L L L L vLR Growth G = L L + kRL

Filtering loss FL = bML

Other losses EL = lLL

Outflow OL = QL Zooplankton, Z dZ = G – E – O dt Z Z Z εaSZ Growth G = S Z ++ 1 aTSSS aTL LL

Losses EZ = lZZ

Outflow OZ = QZ Note: Each component is expressed in terms of the concentration of phosphorus associated with that component. Each process contributing to the overall dynamics of a component is separately defined. Reasonable values for the parameters that govern these processes are listed in Table 2.

Finally, zooplankton density increases with filtering (GZ) and Phytoplankton growth follows the Monod model; for small decreases due to outflow (OZ) and excretion and other mor- phytoplankton, with maximum growth rate vS and half- tality (EZ): saturation concentration kS,

vRSS (4) dZ/dt = GZ – EZ – OZ (5) G = S + RkS The functions in eqs. 1–4 are based on a set of assumptions regarding phosphorus transfer between trophic groups. The expression for GL is equivalent, with subscripts changed First, we assume constant inflow phosphorus concentration appropriately. (Rin) and inflow volume per day (q). If basin volume (V)is We model zooplankton filtering with a type 2 functional constant, inflow and outflow rates are identical. Hence, the response. We assume that large phytoplankton can interfere outflow loss term for each model component is proportional with zooplankton filtering, but that zooplankton can not in- to the concentration of that component in the lake (e.g., OS = gest large phytoplankton. Hence, zooplankton spend time QS, where Q = q/V is identical for all components in the handling large phytoplankton but do not directly affect large model), and inflow of available phosphorus is IR =QRin. phytoplankton dynamics. For attack rates aS and aL on small

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Table 2. Parameter estimates for the western basin of Lake Erie. Parameter Description (units) Value –1 lS Loss rate from edible phytoplankton (day )0.01 –1 lL Loss rate from inedible phytoplankton (day )0.01 –1 lZ Loss rate from zooplankton (day )0.31 ε Zooplankton assimilation efficiency (unitless) 0.54 µ –1 aS Zooplankton attack rate on edible phytoplankton (L·( g P·day) )0.18 µ –1 aL Zooplankton attack rate on inedible phytoplankton (L·( g P·day) ) Variable TS Zooplankton handling time for edible phytoplankton (days) 0.32 TL Zooplankton handling time for inedible phytoplankton (days) 0.32 –1 vS Edible phytoplankton maximum phosphorus uptake rate (day )1.8 µ –1 kS Edible phytoplankton half-saturation phosphorus concentration ( g P·L )0.7 –1 vL Inedible phytoplankton maximum phosphorus uptake rate (day )0.8 µ –1 kL Inedible phytoplankton half-saturation phosphorus concentration ( g P·L )0.7 f Fraction of phosphorus consumed by mussels that is returned to the water column (unitless) 1 µ –1 Rin Inflow phosphorus concentration ( g P·L )33 Q Water outflow rate (day–1)0.02 b Mussel filtering rate (L·day–1) Variable M Mussel density (L–1) Variable Note: See Appendix B for sources. −ε and large phytoplankton, respectively, handling times TS and 1 ε (7) ER = ES + EL + EZ + FZ + fbM(S + L) TL, and assimilation efficiency , ε εaSZ (6) G = S where M is mussel density. A more realistic model structure Z ++ 1 aTSSS aTL LL would include a detritus compartment through which dead and egested material must pass before re-entering the avail- We do not explicitly model losses to higher trophic groups able phosphorus pool. However, we have found through (e.g., fish predation on zooplankton) to keep the model sim- numerical simulations that the addition of a detritus com- ple. However, if the population dynamics of these groups are partment has little effect on our qualitative results, although not tightly linked over the seasonal time scale of plankton the values of the equilibria depend on the rate at which detri- dynamics, such consumption may be treated as part of the tus is recycled. loss terms (e.g., EZ). We make two simplifying assumptions regarding dreissenid Qualitative behaviour filtering. First, we model dreissenid filtering with a type 1 Here, we derive general qualitative predictions for the ini- functional response, with attack rate b. Although dreissenid tial impact of the zebra mussel invasion on equilibrium plank- filtering rate decreases when phytoplankton density exceeds ton densities (Table 3). These predictions are simply the a threshold value (Walz 1978; Sprung and Rose 1988; Berg signs of the derivatives of the equilibria with respect to M, et al. 1996), this threshold is typically greater than the densi- calculated at M = 0. In Appendix A, we derive conditions ties in systems considered here, and explicit consideration of under which the initial impact of dreissenid invasion on the the threshold does not affect our results. Second, we assume > zooplankton equilibrium (Z*) is positive (*/|ddZMM =0 0). dreissenids filter small and large phytoplankton groups indis- If large phytoplankton do not interfere with zooplankton fil- criminately. Although there is evidence of selectivity among tering (aLTL = 0), then this positive effect depends on two phytoplankton species, it generally occurs at the stage of rejec- conditions. First, the parameter combination (kL – kS)(Q + lL) tion in pseudofeces (Ten Winkel and Davids 1982; Horgan must be small relative to the product of large phytoplankton and Mills 1997; Bastviken et al. 1998) after cells and colonies maximum growth rate and small phytoplankton half- have been removed from the water. However, Vanderploeg et saturation concentration (vLkS). The sum Q + lL will typi- al. (2001) found that toxic Microcystis colonies rejected by cally be small in systems such as the western basin of Lake zebra mussels could return to the water column and suggested Erie (inflow is small relative to total basin volume, large that such filtering selectivity could facilitate Microcystis phytoplankton lose little phosphorus; see Appendix B), en- blooms. This situation involves a radical shift in the plank- suring that this condition is met. The second condition is tonic food web not considered in this paper. vk The final set of transfers involves phosphorus recycling. (8) SL>1 For simplicity, we assume that all phosphorus losses from vkLS planktonic groups (excretion, sinking, zooplankton egestion, etc.) return directly to the available phosphorus pool. In con- If small phytoplankton are better phosphorus competitors ≥ trast, we allow a variable fraction (f) of mussel filtrate to be than large phytoplankton (vS > vL, kL kS), the inequality in recycled, whereas the remainder (1 – f) is lost from the sys- eq. 8 must be true. If large phytoplankton interfere with zoo- tem via sedimentation and burial. Given these assumptions, plankton filtering (aLTL > 0), dZ*/dM at M = 0 can increase

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Table 3. Equilibria for the basic model.

All present k() l++ bM Q R*= LL −+ + vLL() l bM Q

()(*)lQ++1 aTL S*= ZLL ε −+ aaTlQSSSZ() −1  ()*1 − fbMS   ()1 − fbM L*=R − −++(*RSZ * *) 1 +   in Q   Q 

 vR*  1 ++aTS** aTL Z*= S −−lbMQ −  SS LL  + S  Rk* S   a S   ()**1 − ε aSZ  L extinct R*=QR+++ l S** l Z S + fbMS* Q−1 in S Z +  1 aTSSS * 

lQ+ S*= Z ε −+ aaTlQSSSZ()  vR*  1 + aTS* Z*= S −−lbMQ −  SS  + S  Rk* S   a S  kbMQ()++ l L and Z extinct R*= SS −++ vbMQlSS() QR(*)− R S*= in vR* S −−lfbM + S kRS * Note: As dreissenid filtering impact increases, the food chain undergoes successive simplifications as first the large phytoplankton and then zooplankton are driven to extinction (see Fig. 1). Equilibrium solutions for these simplified food chains follow the solution for the complete, native food chain. Explicit solutions can be found in each case, but extremely long solutions are left in implicit form.

or decrease with aLTL and possibly become negative if inter- negative effect of dreissenids on zooplankton in some sys- ference is strong relative to handling time for small phyto- tems. If mussel density (or filtering rate, b) is sufficiently plankton. low or the increase in Z* is relatively small, the effect of Direct inspection of the equation for R* (Table 3) shows dreissenids on zooplankton may be difficult to detect in sur- that the equilibrium concentration of available phosphorus vey data. We assess the plausibility of this explanation by always increases with mussel density. Furthermore, we dem- first parameterizing the model. We then ask whether dreis- onstrate in Appendix A that if Z* and R* increase with senid density and filtering rate fall within the range over dreissenids, then S* and L* must decrease. which equilibrium zooplankton density increases and whether We conclude that the initial increase in equilibrium zoo- the predicted increase is smaller than within-season fluctua- plankton concentration with dreissenids does not require any tions or measurement error. of the mechanisms hypothesized to offset mussel consump- We estimated parameter values for the western basin of tion of phytoplankton. Instead, dreissenid filtering alters the Lake Erie from the literature (Table 2). Because this basin is competitive balance between the two phytoplankton groups. shallow and does not stratify, its limnological features may The additional mortality disproportionately affects the weaker more closely approximate a well-mixed compartment than competitor, L, and thereby increases the productivity of ed- most natural systems with zebra mussels. Furthermore, esti- ible phytoplankton. The increased productivity is transferred mates of phosphorus loading, dreissenid abundance, and to zooplankton biomass. This mechanism is independent of plankton density before and after the arrival of dreissenids phosphorus recycling by dreissenids: the parameter f does are available in the literature. not appear in the expression for Z*ifaLTL =0.However,in- We detail the parameterization in Appendix B. We use av- creasing f increases the filtering impact that can be with- erage phosphorus loading (Rin) for 1987–1990, i.e., 2 years stood by the planktonic food web (see below). before and 2 years after zebra mussels established dense populations in the western basin of Lake Erie. Lumping Quantitative behaviour: western basin of Lake Erie phytoplankton into two trophic groups necessitates some sacrifice of precision, and data for phosphorus uptake by Parameterizing the model phytoplankton species separated by edibility to zooplankton The general qualitative predictions of the model offer a are difficult to obtain. Hence, we take the approach sug- potential explanation for the apparent absence of a strong gested by Andersen (1997): we use the upper and lower

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quartiles of the distribution from his survey of measured Fig. 1. Equilibria with parameters estimated for the western basin species values for high and low maximum uptake rates (vS of Lake Erie and no inedible phytoplankton interference (aL =0): and vL). Wu and Culver (1991) found that two cladocerans, (a) solid line, available phosphorus, R*; broken line, large phyto- Daphnia galeata and Daphnia retrocurva, were the domi- plankton, L*; (b) solid line, small phytoplankton, S*; broken line, nant zooplankton grazers during the dreissenid invasion of zooplankton, Z*. For high filtering impact (bM), S* → 0 and R* → the western basin of Lake Erie. Our parameter values for Rin (not shown). Arrows indicate low and high estimates of whole- zooplankton are based on D. galeata. basin filtering impact. Estimates of local filtering impact are too In our analysis, we focus on the occurrence of large and large to be displayed in this plot. consistent deviations between observed and predicted behav- iors and examine the sensitivity of these discrepancies to differences in the structure of our models. We do not present a detailed analysis of sensitivity to parameter variation for any one model structure because the effects produced, for reasonable parameter ranges, are small relative to the size of the deviations that we are interested in accounting for.

Responses of the food chain to increased mussel filtering impact We show the predicted effects of dreissenid filtering on the equilibria for the western basin of Lake Erie (Fig. 1). Here, we use the product of mussel density and filtering rate (bM) as a measure of mussel “filtering impact”. This parameter combination represents the total filtering rate of the mussel population and is equivalent to empirical measurements that incorporate effects of factors such as mussel size structure (e.g., MacIsaac et al. 1992). Filtering impact is equal to the inverse of the time required by the mussel population to filter the entire water column. The predictions of the parameterized model are consistent with the general analysis in the previous section: as bM increases, zooplankton and available phosphorus initially increase and total phytoplankton decreases (L decreases, but S is independent of filtering impact if large phytoplankton do not interfere with zooplankton filtering). Further increases in bM drive large phytoplankton and then zooplankton extinct. To generate Fig. 1, we let f = 1.0, which implies that all phosphorus consumed by dreissenids is returned to the water column via excretion or decomposition of feces, pseudo- rate (Kryger and Riisgård 1988). We calculate two estimates of feces, and mussel biomass. Although complete recycling is filtering impact based on the local density of mussels on reefs unrealistic, decreasing f reduces the filtering impact that can or the average density assuming that the impact is distributed be tolerated by the planktonic food web (all lines in Fig. 1 over the entire basin. move to the left); hence, f = 1.0 gives the most conservative Estimates of zebra mussel filtering rate, measured in the estimate of dreissenid effects. In simulations of the model, laboratory and in the field, also vary considerably (Reeders we found that loss of even a small fraction of phosphorus et al. 1993). Padilla et al. (1996) used a value from Walz consumed by dreissenids has an extremely large negative ef- (1978), which is approximately in the middle of the range of fect on plankton productivity. We return to this issue below. 15 studies reviewed by Reeders et al. (1993, their table 1). Given the density of mussels and their filtering rate, we can MacIsaac et al. (1992) used the allometric scaling function calculate the filtering impact for the western basin of Lake for size-specific filtering rate from Kryger and Riisgård Erie, i.e., the appropriate point on the horizontal axis in Fig. 1. (1988). This function gives the highest value of filtering rate We can then use the model to predict the effect of the dreis- among the estimates compiled by Reeders et al. (1993). We senid invasion on phytoplankton and zooplankton populations. compare the predictions of our model using low (Walz 1978) Zebra mussels primarily colonize hard substrate on reefs in the and high (Kryger and Riisgård 1988) estimates of filtering western basin of Lake Erie. This habitat is patchily distributed rate. These estimates ignore the effects of refiltration (Yu and makes up only approximately 15% of the total surface area and Culver 1999); we consider the effects of incomplete of the basin (MacIsaac et al. 1992). Within these patches, mus- mixing explicitly with the two-compartment model below. sel abundance varies widely, and surveys have found densities The resulting values of local and whole-basin average fil- greater than 3 × 105 mussels·m–2 (Leach 1993). We use popula- tering impact for the two filtering rate measurements are tion survey data from MacIsaac et al. (1992) because their den- listed in Table 4. Locating these points in Fig. 1 shows that sity estimates are typical of samples taken during the first few for each estimate of filtering impact, the model predicts se- years after the dreissenid invasion and they measured size vere impacts on large phytoplankton and zooplankton. For structure, which has a large effect on the estimate of filtering both estimates of local impact, zebra mussel filtering drives

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Table 4. Estimates of filtering impact (bM) for the Fig. 2. (a) Effect of large phytoplankton interference on equilib- western basin of Lake Erie. rium zooplankton concentration, Z*: solid line, aL = 0; broken

–1 line, aL =0.5aS; dotted line, aL = aS. All three lines are identical Filtering impact (day ) for bM >0.75(L* = 0). (b) Effect of self-shading on zooplank- Filtering rate Local Whole basin ton equilibrium. Broken line indicates Z* without self-shading. Low 2.93 0.44 Predictions with and without self-shading are identical for bM < High 13.3 2.0 0.73 (L*>0inthemodel with self-shading). Note: See Appendix B for sources.

zooplankton and large phytoplankton extinct. Using the larger of the averaged values gives the same result, whereas the smaller value results in zooplankton density increasing by an order of magnitude. These predictions are clearly inconsistent with survey data. Samples from the western basin of Lake Erie immediately following the dreissenid invasion show that zooplankton neither increased nor decreased significantly but generally remained within the bounds of pre- dreissenid seasonal variation (Wu and Culver 1991). Although inedible phytoplankton may have decreased relative to edible phytoplankton (Wu and Culver 1991), neither trophic group has been excluded from the community. Wu and Culver’s (1991) data might have been collected before the system reached the predicted equilibrium values; however, more recent data (e.g., Johannsson et al. 2000) from the western basin of Lake Erie confirm that zooplankton have neither increased nor crashed. Model behaviour before the mussel invasion also deviates significantly from observed behaviour. Predicted equilibrium phytoplankton densities in the absence of zebra mussels are roughly an order of magnitude higher than the densities observed just before the zebra mussel invasion: given bM =0 and the parameter values in Table 2, predicted values for R*, S*, L*, and Z* are 0.027, 11, 21, and 0.67 µg P·L–1, respectively; thus the value for S*+L* is approximately eight times the value of 4 µg P·L–1 that Leach (1993) observed in the western basin of Lake Erie in 1988 (conversion from µg chlorophyll (chl)·L–1 to µg P·L–1 assuming the 1:1 P-to-chl ratio given by Reynolds 1984) and is approximately three plankton and zooplankton to mussel filtering is greatly in- times the value of 12 µg P·L–1 summer chlorophyll expected, creased, thus exacerbating the discrepancy between observed given the input phosphorus concentration for our model and predicted behaviour already noted above. µ –1 (Rin =33 g P·L ) and the Dillon–Rigler (Dillon and Rigler We now ask whether additional positive effects on either 1974) statistical relationship linking spring phosphorus con- zooplankton feeding efficiency (i.e., reduction in feeding in- centrations to summer chlorophyll densities. This discrep- terference by inedible zooplankton) or phytoplankton pro- ancy can be easily explained. When bM = 0, our model is ductivity (i.e., reduction of phytoplankton self-shading) structured so that the total phosphorus concentration in the arising from declines in phytoplankton abundance could be water (R*+S*+L*+Z*) must equal the inflow concentra- sufficient to decrease the sensitivity of the zooplankton com- tion (Rin). Although predicted R* and Z* values (0.027 and munity to mussel filtering (i.e., expand the range of filtering 0.67 µg P·L–1, respectively) may be too small compared with impacts that zooplankton can withstand; Fig. 1). pre-dreissenid data (e.g., zooplankton biomass on the order Inedible phytoplankton interference is simply time spent of 100 µg·L–1 dry mass (Wu and Culver 1991) or approxi- by zooplankton handling inedible phytoplankton. In the ba- µ –1 mately 1.8 g P·L , using the conversion factor from Peters sic model, this is represented by aLTL > 0. The effects of in- and Rigler (1973)), altering parameter values to redistribute cluding interference in the model are displayed in Fig. 2a. phosphorus such that S* and L* take on smaller, more rea- We let TL = TS and vary the attack rate, aL.AsaLTL in- sonable values would clearly make the values for R* and Z* creases, the potential for dreissenid filtering to reduce inter- too large. Hence, the most likely explanation for the discrep- ference increases and the initial positive effect of dreissenids ancy between model predictions and data for pre-dreissenid on zooplankton increases. However, including interference invasion phytoplankton densities is that in the real system, in the model does not affect the value of filtering impact at phosphorus accumulates in components (e.g., detritus) that which Z*=0. are not explicitly represented in our well-mixed model. Self-shading reduces the phytoplankton population growth However, if a phosphorus diversion of this sort is introduced terms, GS and GL. We apply the self-shading function used into our model, we find that the sensitivity of both phyto- by Padilla et al. (1996), which was described by Carpenter

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(1992). This function reduces GS and GL in proportion to the Fig. 3. Consumption rate per individual mussel at equilibrium total phytoplankton density, S + L. For small phytoplankton, phytoplankton density, using the lower estimate of filtering rate, b = 1.5 L·day–1 (Walz 1978).  vSR  SL+  (9) G =  S  1 −  S +    kRS  z

where z =97µg P·L–1 for the western basin of Lake Erie (8 m average depth; see Carpenter 1992). The expression for GL is also multiplied by the term1–(S + L)/z. As mussel filtering impact increases, phytoplankton density decreases, and this reduces self-shading. However, this modification of the basic model has only minor quantitative effects on the predictions for Z* (Fig. 2b). Over the range of filtering impact for which L* > 0, the predictions for Z* are identical. In this region, self-shading decreases large phytoplankton density, but small phytoplankton benefit from higher concentrations of available phosphorus. This compen- sation for the reduction of primary production by self- shading maintains zooplankton at the same density whether the self-shading effect is present or absent in the model. Once the filtering impact exceeds the level sufficient to drive the large phytoplankton to extinction, we see a relatively rates could stem from overestimating the mussel filtering small reduction in the productivity of the small phyto- rate (b). However, laboratory-based estimates of b are more plankton that remain. This is reflected in a small reduction likely to underestimate the true value, not overestimate it, in equilibrium zooplankton abundance and, consequently, re- because they are derived from mussels held in an unnatural duction in the filtering impact sufficient to drive zooplankton environment. Furthermore, dreissenids are more likely to fil- to extinction. ter near their maximum rate (Walz 1978; Sprung and Rose 1988) when they are exposed to low food densities. Hence, Predicted mussel ingestion rates and expected mussel we suspect the problem lies elsewhere. food requirements Our results suggest that the dreissenid population in the Failures of the well-mixed model: a summary western basin of Lake Erie is capable of consuming primary The responses of our well-mixed model to a zebra mussel production rapidly enough to drive zooplankton and even invasion failed to match many of the observed responses of some phytoplankton taxa extinct, yet such extinctions have the western basin of Lake Erie to the mussel invasion that not been observed. A possible explanation for this discrep- began there in 1989. Some of the more instructive failures ancy between model predictions and field observations is are listed here. (i) The model predicts that zooplankton that gross overestimates of mussel ingestion rates are ob- abundance will increase significantly in the early stages of tained when measured mussel filtering rates are simply mul- the invasion, when overall filtering rate is still relatively low; tiplied by average water column phytoplankton densities. To this was not observed. (ii) The model predicts that both zoo- examine this possibility, we compared mussel ingestion rates plankton and larger species of phytoplankton will be driven predicted using this approach with an estimate of the food to extinction by mussel filtering rates that are significantly requirements of a typical mussel, as derived from the dreis- less than those currently operating in the real system; this senid energy budget developed by Stoeckmann and Garton was not observed. (iii) Realistic phosphorus input rates pro- (1997). This energy budget is based on laboratory measure- duce equilibrium phytoplankton densities in the model that ments of respiration plus field estimates of total growth and are an order of magnitude greater than observed; this dis- reproductive investment in the western basin of Lake Erie. It crepancy can be remedied by allowing some phosphorus to provides an estimated daily ration of 3.85 cal·day–1 for an accumulate in a detritus pool; however, diverting phosphorus individual mussel in the field and this converts to 0.15 µg into detritus (e.g., f < 1) further reduces the maximum filter- P·day–1, using the following conversion factors: 1000 cal·g–1 ing impact that large phytoplankton and zooplankton can phytoplankton wet weight (Stoeckmann and Garton 1997), withstand. (iv) Mussel consumption rates derived from ob- 0.1 g C·g wet weight–1 (Shuter 1978), and the C–P ratio of served values for both mussel filtering rate and phyto- 48:1 in phytoplankton biomass (Reynolds 1984). plankton density are much larger than expected mussel food The ingestion rate estimate obtained by multiplying mea- requirements. sured filtration rates (approximately 1.5 L·day–1 (Walz 1978)) Each of these discrepancies ultimately derives from the and Leach’s (1993) post-dreissenid invasion phytoplankton assumption that all of the phytoplankton in the water column abundance measurements (monthly mean chlorophyll for are equally accessible to zebra mussel predation. This as- 1989–1990 varies from 1 and 3 µg·L–1) is 1.5–4.5 µg P·day–1, sumption denies a basic fact of zebra mussel biology, an order of magnitude larger than the bioenergetics estimate namely that they are bottom dwellers, whose access to sur- of mussel food requirements. If we compare estimated con- face water is dependent on high levels of turbulence and the sumption rates (= b(S*+L*)) from our mixed model over a absence of bottom boundary layers. In the section that fol- range of mussel densities (Fig. 3), we obtain ingestion rates lows, we extend our model to explicitly recognize the spatial that are even higher. These unrealistically high ingestion separation between mussels and their prey, and we show that

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Table 5. Equations for the two-compartment model.

dR ()1 − ε aSZ vSR vLR 1 =QR() −++++ R lS lL lZ S 11 +−−µ ()RR S 11− L 11 in11SLZ 1 1 + 12 1 + + d1+t aTSSS1 aTLLL1 Rk1 S Rk1 L

dR ()1 − ε aSZ vSR vLR 2 =QR() −++++ R lS lL lZ S 22 +−−µ ()RR S 22− L 22++fbM ( S L ) in22SLZ 2 2 + 21 2 + + 22 d1+t aTSSS 2 aTLLL 2 Rk2 S Rk2 L

dS vSR  aSZ  1 = S 11−+()QlS − S 11  +−µ ()SS + S 1 ++ 12 1 d1t Rk1 S  aTSSS11 aTL LL 

dS vSR  aSZ  2 = S 22−++−()bM Q l S  S 22  +−µ ()SS + S 2 ++ 21 2 d1t Rk2 S  aTSSS2 aTL LL2 

dL vLR 1 = L 11 −+()()Ql +µ L − L + L 12 1 dt Rk1 L

dL vLR 2 = L 22−+++−()()bM Q lµ L L + L 21 2 dt Rk2 L

dZ εaSZ 1 = S 11 −+()lQZZZ +µ ( − ) + Z 1121 d1+t aTSSS11 aTL LL

dZ εaSZ 2 = S 22 −+()lQZ +µ ( ZZ − ) + Z 2212 d1+t aTSSS22 aTL LL

this extension permits a simple resolution of all of the dis- ture, dreissenids are potentially limited by the rate of delivery crepancies between predicted and observed behaviour that of phytoplankton from the unoccupied to the occupied com- we identified in our well-mixed model. partment. The two-compartment structure is not intended to represent true compartmentalization of lakes, such as ther- mal stratification, which severely restricts mixing. Rather, it Model II: partial mixing in a two-compartment is the simplest approximation to a continuously mixing basin model in which localized mussel populations create differences in the effective filtering impact across space. Dreissenids are restricted to the bottom surface of the basin and potentially refilter the lowest portion of the water Equations for the two-compartment model are listed in column more quickly than it mixes with water near the sur- Table 5. Modifications consist of splitting each food web face (Yu and Culver 1999; Ackerman et al. 2001). This hy- component into populations in compartments occupied and pothesis is supported by chlorophyll samples taken at unoccupied by zebra mussels and adding a mixing term to multiple depths above mussel beds that show a gradient of each equation. For simplicity, we assume that all four com- increasing concentration with height for approximately 1– ponents have equal mixing rates between compartments and 2 m above the benthos (MacIsaac et al. 1992, 1999; Ackerman that basin inflow and outflow enter and leave the compartments in proportion to their relative volumes. The mixing et al. 2001). Although dreissenid filtering does not necessar- µ ily cause this gradient, low prey availability near the benthos rate for compartment i ( i) is proportional to the rate of implies that mussel filtering impact may be limited by the water crossing the boundary surface shared by the two com- rate of vertical mixing. Furthermore, the patchy distribution partments, divided by the volume of the compartment: of dreissenid populations within a basin suggests that mussel σ filtering impact may also be limited by horizontal mixing. µ = A (10) i These considerations suggest that there may be effective spa- V tial separation between zebra mussels and their prey. We ex- i tend our well-mixed model to explicitly include such spatial separation in order to examine its role in moderating the im- where A is the shared surface area, Vi is the volume of com- pact of zebra mussel filtering on the pelagic food chain. partment i, and σ is a rate coefficient (with units m·day–1). The three parameters in eq. 10 are functions of physical Structure characteristics of the basin that are difficult to quantify. To analyze the effects of spatial structure on the predicted Shared surface area and volume of each compartment de- impact of mussels, we incorporate the simplest possible rep- pend on the fraction of the bottom of the basin that is occu- resentation of spatial processes in our model. We divide the pied by dreissenids, the patchiness of the occupied habitat, well-mixed compartment into two compartments, one occu- and the fraction of the water column that is included in the pied by dreissenids and the other unoccupied. Mussel filter- occupied compartment. The mixing rate coefficient σ de- ing directly affects only the occupied compartment, but the pends on turbulence in the water column generated by forces two compartments are connected by mixing. In this struc- such as wind or currents.

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Fig. 4. Equilibria for the two-compartment model as functions of water mixing rate: (a) zooplankton; (b) phosphorus; (c) small phytoplankton; and (d) large phytoplankton. We arbitrarily assume that the occupied compartment includes one-half of the water column. Relative size of the two compartments is held constant by increasing the mixing rates by the same factor, maintaining the µ µ –1 ratio 2/ 1 = V1/V2 = (1 – (0.5)(0.15)/((0.5)(0.15)) = 12.3 for 15% occupied surface. We set bM =5.86day , the lower value of the local filtering impact multiplied by 2 in order to maintain the same mussel abundance used in the original calculation of filtering im- –1 ᭺ pact. The whole-basin average filtering impact is therefore bM = 5.86(V2/(V1 + V2))=0.44day . , compartment 1, mussels absent; ×, compartment 2, mussels present. Arrows and dotted lines indicate basic single-compartment model equilibria for (x) mussels absent, (y) bM = 5.86 day–1, and (z) bM =0.44day–1.

Qualitative behaviour mussel filtering impact is sufficient to drive all three plank- A thorough analysis of the effects of spatial structure on ton groups extinct. In the unoccupied compartment, the the model behavior is beyond the scope of this paper. Here, equilibria approach values from the single-compartment we present numerical solutions for the equilibria over a lim- model with bM = 0. As mixing increases, the two compart- ited range of parameter values in order to show how mussel ments act increasingly like a single well-mixed basin with filtering can suppress phytoplankton populations, despite mussel filtering impact diluted by the additional water vol- having a negligible effect on zooplankon abundance. In par- ume of the unoccupied compartment. ticular, we examine two abiotic features that are clearly We demonstrate the responses of the equilibria to increas- µ linked to mussel filtering impact. First, increasing both is ing the area of the occupied compartment with filtering im- by the same factor increases the rate of mixing between the pact held constant (Fig. 5). Increasing the fraction of the two compartments caused by, for example, increasing turbu- benthos occupied by dreissenids implies that V2 increases lence (higher σ) or more patchily distributed mussel beds and V decreases. If we assume that mixing increases with µ µ 1 (higher A). Second, increasing 1/ 2 = V2/V1, where i =1 the proportion of benthic surface area occupied by for the unoccupied compartment and i = 2 for the occupied dreissenids (i.e., vertical mixing), the ratio A/V2, and there- µ compartment, represents increasing the fraction of the bot- fore 2, are constant (eq. 10). Similarly, V1 decreases and A µ tom that is occupied or the fraction of the water column that increases, so 1 increases with occupied surface area. We fix µ is included in the occupied compartment, either of which in- 2 at a low value, where the equilibria approach values for creases the volume of the occupied compartment (V2). isolated basins in Fig. 4, so spatial structure has a large ef- µ Equilibria calculated over a range of mixing rates with fect. (For large 2, the basin is approximately well mixed, µ µ constant 1/ 2 are displayed (Fig. 4). The equilibria vary be- and increasing the fraction of bottom surface occupied is ef- tween two extremes. First, with low mixing, the two com- fectively the same as increasing bM in Fig. 4.) The results in partments act approximately like isolated basins, either with Fig. 5 demonstrate how increasing the fraction occupied by or without dreissenids. Hence, the equilibria approach values mussels enriches the unoccupied compartment with recycled calculated from the single-compartment model. In the occu- phosphorus but simultaneously increases phytoplankton pied compartment, S*, L*, and Z* approach zero because the losses to mussel filtering in the occupied compartment. The

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µ µ µ –1 Fig. 5. Equilibria for the two-compartment model with increasing fraction of surface occupied (increasing 1/ 2, 2 = 0.1 day ): (a) zooplankton; (b) phosphorus; (c) small phytoplankton; and (d) large phytoplankton. In the occupied compartment, bM =5.86day–1 (see Fig. 4 legend). ᭺, compartment 1, mussels absent; ×, compartment 2, mussels present. Equilibrium phosphorus concentration in compartment 1 multiplied by 100 for clarity.

changing balance between nutrient delivery and population fraction of phosphorus returned to the water column (f = losses favors the faster-growing small phytoplankton, which 0.01; Fig. 6). Under these conditions, zooplankton density in supports higher equilibrium zooplankton density. the unoccupied compartment is insensitive to mussel filter- We show that spatial isolation of mussels allows zoo- ing impact, whereas both phytoplankton groups decrease but plankton and phytoplankton to persist, despite high mussel persist for a broad range of plausible values. Most phospho- density and filtering rate, by reducing the effective filtering rus consumed by mussels is lost from the system, and the impact (Figs. 4, 5). However, spatial isolation reduces the competitive advantage of small phytoplankton over large positive effect of mussel filtering on zooplankton only phytoplankton is offset by reduced nutrient supply. slightly and only if mixing is low enough that the two compartments act as nearly separate basins. With such slow mixing, the effect of mussel filtering on both phytoplankton Discussion groups is weak (Fig. 4). Even at this low mixing rate, the zooplankton equilibrium increases with the fraction of ben- We used a simple model to show that basic assumptions thic surface that is occupied by mussels, but the effect on about plankton dynamics result in unrealistically extreme ef- phytoplankton abundance is minimal (Fig. 5). Indeed, the fects of dreissenid filtering. However, consideration of the mechanism causing the increase in zooplankton in the single- spatial structure created when sessile benthic filter feeders compartment model operates in the identical fashion in the invade a pelagic food web leads to radically different results, two-compartment model: mussel filtering decreases the com- which are consistent with empirical data from the western petitive effect of large phytoplankton, which allows small basin of Lake Erie. Previous studies have suggested that phytoplankton to acquire a larger share of the available dreissenid filtering impact is not simply related to mussel phosphorus and indirectly benefits zooplankton. density, as in the well-mixed case, but also depends on local The importance of nutrient competition to predicting the effects. For instance, Yu and Culver (1999) demonstrated effect of dreissenid filtering on zooplankton suggests that that mussels refilter water more rapidly than seston is re- sequestration and burial of phosphorus by mussels provides plenished by mixing and that refiltering increases with mus- a key to understanding the insensitivity of zooplankton pop- sel clumping. MacIsaac et al. (1999) reproduced the observed ulations to the dreissenid invasion in the western basin of vertical gradient in chlorophyll concentration over a mussel Lake Erie. Although reducing the fraction of phosphorus re- bed with a simulation model of mussel grazing and phyto- turned to the water column by mussels quickly reduces pro- plankton distribution in a continuous vertical cross section of ductivity such that the planktonic food web cannot persist in the water column. Here, we simplified the spatial structure the single-compartment model, spatial isolation can mini- into distinct compartments occupied and unoccupied by mize this effect of dreissenids. We demonstrate the effect of dreissenids to analyze the indirect effects of mussel filtering dreissenid filtering on the equilibria for low mixing and low on zooplankton and phytoplankton.

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1364 Can. J. Fish. Aquat. Sci. Vol. 60, 2003

Fig. 6. Equilibria for the two-compartment model with increasing filtering impact, low phosphorus recycling by mussels (f = 0.01), and µ –1 µ –1 low mixing ( 1 = 0.01 day , 2 = 0.123 day ): (a) zooplankton; (b) phosphorus; (c) small phytoplankton; and (d) large phytoplankton. ᭺, compartment 1, mussels absent; ×, compartment 2, mussels present.

Our analysis of the single-compartment model, in which proportional to their density in the basin. This assumption is we assumed the entire water column is well mixed, demon- based on the expectation that the biotic and abiotic compo- strated that the initial positive effect of dreissenids on zoo- nents of the model will exhibit largely passive movement plankton density is predicted for a broad range of conditions. with respect to water outflow from the system. In contrast, This response is driven by the difference in competitive abil- Padilla et al. (1996) assumed that biotic components remain ities of the two phytoplankton groups. Such differences have in the system, whereas available phosphorus is flushed out at been documented in diverse taxa (reviewed in Andersen 1997). a constant rate, independent of its concentration. If the flush- By excluding terms describing phytoplankton self-shading ing rate is less than the inflow rate, the system accumulates and inedible phytoplankton interference with zooplankton phosphorus with time. It is difficult to see how this structure filtering, we showed that these mechanisms can contribute to could be realized in the field; however, it is this buildup of the positive effect, but they are not necessary to explain it. phosphorus over the growing season that offsets phyto- However, the initial increase in zooplankton is not sufficient plankton production lost to zebra mussel filtering and thus is to offset high mussel filtering impact and the model predicts fundamental to the prediction of the Padilla et al. (1996) that zooplankton disappear from the system at filtering rates model that zooplankton will be relatively insensitive to zebra that are much lower than those that have been observed. In- mussel filtering. In our model, the same prediction derives deed, with a well-mixed water column, the food web cannot from the well-founded assumption that the phytoplankton withstand a mussel filtering impact greater than the maxi- community consists of edible and inedible forms of algae mum growth rate of edible phytoplankton, and the value of that compete for the same limiting nutrient. In our well- this parameter (vS) is bounded by a large body of empirical mixed model, careful accounting for phosphorus inflow and data (e.g., Andersen 1997). We conclude that even with more outflow leads to the prediction that the zooplankton popula- careful parameterization, our well-mixed, single-compartment tion increases for low mussel abundance but can not with- model can not account for the observed insensitivity of zoo- stand realistically high mussel filtering impact. plankton abundance to mussel filtering. However, our model is still a highly simplified representa- The well-mixed model developed in this paper differs from tion of the plankton community. For instance, in experiments earlier work by Padilla et al. (1996) in several ways. The with two zooplankton taxa with different feeding modes, most important difference is in our treatment of water flow Sommer et al. (2001) demonstrated an interactive effect on through the system and its effect on nutrients and plankton total phytoplankton density. Similarly, the phytoplankton groups. We assumed that water movement through the sys- community and its responses to herbivory and nutrient sup- tem imposes a density-independent loss term on each com- ply can be complex and vary spatially and temporally. Evi- ponent of the food web such that nutrients and all three dence from the western basin of Lake Erie suggests that, in planktonic groups flow out of the system at a rate that is offshore waters, the density of some phytoplankton taxa in-

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creased after dreissenid invasion, whereas others decreased column, even in a shallow, unstratified basin. Alternatively, (Makarewicz et al. 1999; Munawar and Munawar 1999). On zebra mussels may obtain a substantial portion of their food the other hand, Nicholls and Hopkins (1993) found that all from dead or sinking phytoplankton (Yu and Culver 1999) species decreased in nearshore samples. and even dissolved organic carbon (Roditi et al. 2000), Nevertheless, we suggest that the mechanisms underlying which could weaken the link between mussel and phyto- the indirect effects of mussel filtering on plankton popula- plankton dynamics. Knowledge of the rate at which phyto- tions, revealed by our analysis of the basic model, operate in plankton are delivered to the benthos by turbulent mixing natural systems. Indeed, the predicted increase in relative would allow estimation of the relative importance of food abundance of edible phytoplankton in response to mussel fil- and suitable benthic habitat to dressenid population regulation. tering is consistent with data from the western basin of Lake Erie (Wu and Culver 1991). Furthermore, the processes that Acknowledgements drive the counterintuitive effect of dreissenid filtering on zooplankton in the basic model are crucial to explaining our We thank O. Johannsson, S. Peacor, and three anonymous results from the spatially structured model. The ability of reviewers for helpful comments on the manuscript. Funding zooplankton to persist despite strong mussel filtering impact was provided by the Great Lakes Fisheries Commission. depends on limited delivery of phytoplankton to the benthos and phosphorus burial by dreissenids, in addition to differences in edibility and competitive ability between phyto- References plankton groups. Low mixing between compartments reduces Ackerman, J.D., Loewen, M.R., and Hamblin, P.F. 2001. Benthic– the capture rate of mussels, similar to low filtering impact in pelagic coupling over a zebra mussel reef in western Lake Erie. the well-mixed model; however, the positive effect of low Limnol. Oceanogr. 46: 892–904. filtering impact on zooplankton is offset by phosphorus burial Andersen, T. 1997. Pelagic nutrient cycles: herbivores as sources in the two-compartment model. These mechanisms are not and sinks. Springer-Verlag, Berlin. specific to the two-compartment spatial structure that we Bailey, R.C., Grapentine, L., Stewart, T.J., Schaner, T., Chase, M.E., have used in our model but are applicable to a continuously Mitchell, J.S., and Coulas, R.A. 1999. Dreissenidae in Lake On- mixing water column. tario: impact assessment at the whole lake and Bay of Quinte spa- Results from the two-compartment model support the con- tial scales. J. Gt. Lakes Res. 25: 482–491. clusion that water column turnover time, a frequently re- Bastviken, D.T.E., Caraco, N.F., and Cole, J.J. 1998. Experimental ported measure of filtering impact (e.g., MacIsaac et al. measurements of zebra mussel (Dreissena polymorpha) impacts 1992; Bunt et al. 1993; Bailey et al. 1999) that is based on on phytoplankton community composition. Freshw. Biol. 39: the assumption that the water column is well-mixed, gener- 375–386. ally overestimates the direct effect of mussels on plankton Berg, D.J., Fisher, S.W., and Landrum, P.F. 1996. Clearance and dynamics (Yu and Culver 1999; Ackerman et al. 2001). In- processing of algal particles by zebra mussels (Dreissena polymorpha). J. Gt. Lakes Res. 22: 779–788. stead, we suggest that the realized dreissenid filtering impact Berkman, P.A., Haltuch, M.A., Tichich, E., Garton, D.W., Kennedy, is modified by abiotic features of the basin, such as turbulent G.W., Gannon, J.E., Mackey, S.D., Fuller, J.A., and Liebenthal, mixing, depth, and spatial arrangement of suitable benthic D.L. 1998. Zebra mussels invade Lake Erie muds. Nature (Lond.), substrate. These characteristics determine the rate of phyto- 393: 27–28. plankton delivery to mussels. Bridgeman, T.B., Fahnenstiel, G.L., Lang, G.A., and Nalepa, T.F. Mixing rate and phosphorus recycling are probably corre- 1995. Zooplankton grazing during the zebra mussel (Dreissena lated and typically low in the western basin of Lake Erie. polymorpha) colonization of Saginaw Bay, Lake Huron. J. Gt. Turbulent mixing of the water column resuspends mussel fe- Lakes Res. 21: 567–573. ces and pseudofeces, in which much of the captured food is Bunt, C.M., MacIsaac, H.J., and Sprules, W.G. 1993. Pumping deposited (e.g., Walz (1978) found that over 60% of cap- rates and projected filtering impacts of juvenile zebra mussels tured phytoplankton was rejected as pseudofeces at high (Dreissena polymorpha) in western Lake Erie. Can. J. Fish. food concentration). Although wind at the water surface can Aquat. Sci. 50: 1017–1022. mix the entire water column in such a shallow basin, current Carpenter, S.R. 1992. Destabilization of planktonic ecosystems and velocity near the benthos is often low (MacIsaac et al. 1999; blooms of blue-green algae. In Food web management: a case Ackerman et al. 2001). The chlorophyll gradient above mus- study of Lake Mendota. Edited by J.F. Kitchell. Springer-Verlag, sel beds observed by MacIsaac et al. (1992, 1999) and Berlin. pp. 461–481. Ackerman et al. (2001) suggests a combination of filtering at Coakley, J.P., Brown, G.R., Ioannou, S.E., and Charlton, M.N. 1997. the bottom surface and relatively slow turbulent mixing with Colonization patterns and densities of zebra mussel Dreissena in upper portions of the water column. muddy offshore sediments of western Lake Erie, Canada. Water Air Soil Pollut. 99: 623–632. The two-compartment model predicts that if mixing and Czech, B., and Krausman, P.R. 1997. Distribution and causation of dreissenid recycling are low, changes in mussel abundance species endangerment in the United States. Science (Wash., D.C.), above a relatively low density will have little effect on 277: 1116–1117. phytoplankton or zooplankton populations. After the inva- Dermott, R., and Munawar, M. 1993. Invasion of Lake Erie off- sion has surpassed this threshold, mussels capture nearly all shore sediments by Dreissena, and its ecological implications. phytoplankton that enter the occupied compartment. This Can. J. Fish. Aquat. Sci. 50: 2298–2304. prediction implies that dreissenids can be food-limited while Dillon, P.J., and Rigler, F.H. 1974. The phosphorus–chlorophyll re- their food is still abundant in the upper portion of the water lationship in lakes. Limnol. Oceanogr. 19: 767–773.

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Hebert, P.D.N., Muncaster, B.W., and Mackie, G.L. 1989. Ecological Nalepa, T.F., Fahnenstiel, G.L., and Johengen, T.H. 1999. Impacts of and genetic studies on Dreissena polymorpha (Pallas): a new mol- the zebra mussel (Dreissena polymorpha) on water quality: a case lusc in the Great Lakes. Can. J. Fish. Aquat. Sci. 46: 1587–1591. study in Saginaw Bay, Lake Huron. In Nonindigenous freshwater Horgan, M.J., and Mills, E.L. 1997. Clearance rates and filtering organisms: vectors, biology, and impacts. Edited by R. Claudi and activity of zebra mussel Dreissena polymorpha: implications for J.H. Leach. Lewis Publishers, Boca Raton, Fla. pp. 255–271. freshwater lakes. Can. J. Fish. Aquat. Sci. 54: 249–255. Nicholls, K.H., and Hopkins, G.J. 1993. Recent changes in Lake Idrisi, N., Mills, E.L., Rudstam, L.G., and Stewart, D.J. 2001. Im- Erie (North Shore) phytoplankton: cumulative impacts of phos- pact of zebra mussels (Dreissena polymorpha) on the pelagic phorus loadings and the zebra mussel introduction. J. Gt. Lakes lower trophic levels of Oneida Lake, New York. Can. J. Fish. Res. 19: 637–647. Aquat. Sci. 58: 1430–1441. Olsen, Y., and Østgaard, K. 1985. Estimating release rates of phos- Johannsson, O.E., Dermott, R., Graham, D.M., Dahl, J.A., Millard, phorus from zooplankton: model and experimental verification. E.S., Myles, D.D., and LeBlanc, J. 2000. Benthic and pelagic sec- Limnol. Oceanogr. 30: 844–852. ondary production in Lake Erie after the invasion of Dreissena Padilla, D.K., Adolph, S.C., Cottingham, K.L., and Schneider, D.W. spp. with implications for fish production. J. Gt. Lakes Res. 26: 1996. Predicting the consequences of dreissenid mussels on a pe- 31–54. lagic food web. Ecol. Model. 85: 129–144. Johnson, L.E., and Carlton, J.T. 1996. Post-establishment spread in Parker, I.M., Simberloff, D., Lonsdale, W.M., Goodell, K., Wonham, large-scale invasions: dispersal mechanisms of the zebra mussel M., Kareiva, P.M., Williamson, M.H., VonHolle, B., Moyle, P.B., Dreissena polymorpha. Ecology, 77: 1686–1690. Byers, J.E., and Goldwasser, L. 1999. Impact: toward a frame- Kryger, J., and Riisgård, H.U. 1988. Filtration rate capacities in 6 work for understanding the ecological effects of invaders. Biol. species of European freshwater bivalves. Oecologia, 77: 34–38. Invas. 1: 3–19. Leach, J.H. 1993. Impacts of the zebra mussel (Dreissena poly- Peters, R.H., and Rigler, F.H. 1973. Phosphorus release by Daph- morpha) on water quality and fish spawning reefs in western nia. Limnol. Oceanogr. 18: 821–839. Lake Erie. In Zebra mussels: biology, impacts: and control. Pimentel, D., Lach, L., Zuniga, R., and Morrison, D. 2000. Envi- Edited by T.F. Nalepa and D.W. Schloesser. Lewis Publishers, ronmental and economic costs of nonindigenous species in the Boca Raton, Fla. pp. 381–397. United States. BioScience, 50: 53–65. Reeders, H.H., bij de Vaate, A., and Noordhuis, R. 1993. Potential MacIsaac, H.J., Johannsson, O.E., Ye, J., Sprules, W.G., Leach, of the zebra mussel (Dreissena polymorpha) for water quality J.H., McCorquodale, J.A., and Grigorovich, I.A. 1999. Filtering management. In Zebra mussels: biology, impacts, and control. impacts of an introduced bivalve (Dreissena polymorpha)ina Edited by T.F. Nalepa and D.W. Schloesser. Lewis Publishers, shallow lake: application of a hydrodynamic model. Ecosys- Boca Raton, Fla. pp. 439–451. tems, 2: 338–350. Reynolds, C.S. 1984. The ecology of freshwater phytoplankton. MacIsaac, H.J., Sprules, W.G., Johannsson, O.E., and Leach, J.H. Cambridge University Press, Cambridge. 1992. Filtering impacts of larval and sessile zebra mussels Roditi, H.A., Fisher, N.S., and Sanudo-Wilhelmy, S.A. 2000. Up- (Dreissena polymorpha) in western Lake Erie. Oecologia, 92: take of dissolved organic carbon and trace elements by zebra 30–39. mussels. Nature (Lond.), 407: 78–80. Mack, R.N., Simberloff, D., Lonsdale, W.M., Evans, H., Clout, M., Scavia, D., Lang, G.A., and Kitchell, J.F. 1988. Dynamics of Lake and Bazzaz, F.A. 2000. Biological invasions: causes, epidemiol- Michigan plankton: a model evaluation of nutrient loading, com- ogy, global consequences, and control. Ecol. Appl. 10: 689–710. petition, and predation. Can. J. Fish. Aquat. Sci. 45: 165–177. Makarewicz, J.C., Lewis, T.W., and Bertram, P. 1999. Phytoplankton Shuter, B.J. 1978. Size dependence of phosphorus and nitrogen sub- composition and biomass in the offshore waters of Lake Erie: sistence quotas in unicellular microorganisms. Limnol. Oceanogr. pre- and post-Dreissena introduction (1983–1993). J. Gt. Lakes 23: 1248–1255. Res. 25: 135–148. Sommer, U., Sommer, F., Santer, B., Jamieson, C., Boersma, M., McCauley, E., Nisbet, R.M., De Roos, A.M., Murdoch, W.W., and Becker, C., and Hansen, T. 2001. Complementary impact of Gurney, W.S.C. 1996. Structured population models of herbivo- copepods and cladocerans on phytoplankton. Ecol. Lett. 4: 545– rous zooplankton. Ecol. Monogr. 66: 479–501. 550. Mellina, E., and Rasmussen, J.B. 1994. Patterns in the distribution Sprung, M., and Rose, U. 1988. Influence of food size and food and abundance of zebra mussel (Dreissena polymorpha)inrivers quality on the feeding of the mussel Dreissena polymorpha. and lakes in relation to substrate and other physicochemical fac- Oecologia, 77: 526–532. tors. Can. J. Fish. Aquat. Sci. 51: 1024–1036. Stoeckmann, A.M., and Garton, D.W. 1997. A seasonal energy Mellina, E., Rasmussen, J.B., and Mills, E.L. 1995. Impact of ze- budget for zebra mussels (Dreissena polymorpha) in western bra mussel (Dreissena polymorpha) on phosphorus cycling and Lake Erie. Can. J. Fish. Aquat. Sci. 54: 2743–2751. chlorophyll in lakes. Can. J. Fish. Aquat. Sci. 52: 2553–2573. Strayer, D.L., Caraco, N.F., Cole, J.J., Findlay, S., and Pace, M.L. Munawar, M., and Munawar, I.F. 1999. The changing phytoplankton 1999. Transformation of freshwater ecosystems by bivalves. Bio- biomass and its composition in Lake Erie: a lakewide comparative Science, 49: 19–27. analysis. In State of Lake Erie (SOLE): past, present and future. Ten Winkel, E.H., and Davids, C. 1982. Food selection by Dreissena Edited by M. Munawar, T. Edsall, and I.F. Munawar. Backhuys polymorpha Pallas (Mollusca: Bivalvia). Freshw. Biol. 12:553– Publishers, Leiden. pp. 125–154. 558. Nalepa, T.F., and Schloesser, D.W. (Editors). 1993. Zebra mussels: Vanderploeg, H.A., Liebig, J.R., Carmichael, W.W., Agy, M.A., biology, impacts, and control. Lewis Publishers, Boca Raton, Johengen, T.H., Fahnenstiel, G.L., and Nalepa, T.F. 2001. Zebra Fla. mussel (Dreissena polymorpha) selective filtration promoted Nalepa, T.F., Wojcik, J.A., Fanslow, D.L., and Lang, G.A. 1995. toxic Microcystis blooms in Saginaw Bay (Lake Huron) and Initial colonization of the zebra mussel (Dreissena polymorpha) Lake Erie. Can. J. Fish. Aquat. Sci. 58: 1208–1221. in Saginaw Bay, Lake Huron: population recruitment, density, Walz, N. 1978. The energy balance of the freshwater mussel and size strucutre. J. Gt. Lakes Res. 21: 417–434. Dreissena polymorpha Pallas in laboratory experiments and

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Lake Constance. I. Pattern of activity, feeding and assimilation total phosphorus in the system remains the same or de- efficiency. Arch. Hydrobiol. 55: 83–105. creases (if f < 1) with M. Wu, L., and Culver, D.A. 1991. Zooplankton grazing and phytoplankton abundance: an assessment before and after invasion of Dreissena polymorpha. J. Gt. Lakes Res. 17: 425–436. Appendix B: parameterization Yu, N., and Culver, D.A. 1999. Estimating the effective clearance Water inflow and phosphorus concentration rate and refiltration by zebra mussels, Dreissena polymorpha,in Mellina et al. (1995, their table 1) summarize hydrologi- a stratified reservoir. Freshw. Biol. 41: 481–492. cal parameters for the western basin of Lake Erie. To calculate the water inflow rate, we divide average daily water Appendix A: response of equilibria to dreissenid inflow (0.518 km3·day–1) by basin volume (28 km3)toob- invasion tain Q = 0.02 day–1. For the phosphorus concentration of water inflow, we divide the average phosphorus load for First, we differentiate R* with respect to M: 1987–1990 (Nicholls and Hopkins 1993, their table 1) by the dR* kbv average annual water inflow from Mellina et al. (1995). The (A1) = LL µ –1 −+ +2 result is Rin =33 g P·L . dM ((vLL l bM Q )) This expression is positive for all M ≥ 0, so mussel invasion Phytoplankton increases the equilibrium concentration of available phos- The Monod model of phosphorus uptake by phytoplankton phorus. Similarly, depends on two parameters: the maximum uptake rate (vS, vL) and the half-saturation concentration (kS, kL). Based on a dS*()lQaT+ dL* (A2) = ZLL literature survey of parameter estimates for individual phyto- ε −+ dM aTlQSSZ(())dM plankton species, Andersen (1997) suggested that the upper and lower quartiles of the distribution are reasonable approx- ε Because – TS(lZ + Q) > 0 for S* > 0 (Table 3), the change imations for two species groups with a trade-off between in the two phytoplankton groups must have the same sign. competitive ability and grazer resistance. Scavia et al. (1988) Differentiating Z* with respect to M and setting M =0 and Padilla et al. (1996) present values for phytoplankton gives groups distinguished by their edibility to zooplankton in Saginaw Bay (Lake Huron) and Green Bay (Lake Michigan),  dR*   vk  respectively. Estimates of maximum uptake rates for edible dZ* SS 1 ++aTL* aTL* (A3) =  dM − b SS LL  and inedible phytoplankton in both bays are roughly equal to + 2 dM M =0 (*)kRS  a S  the upper and lower quartiles in Andersen’s (1997) survey   distribution. We use the upper quartile for small phyto- –1 plankton (vS = 1.8 day ) and the lower quartile for large  dS**dL  –1  aT + aT  phytoplankton (vL = 0.8 day ).  vR*  SS LL +  S −+()lQ dM dM  The difference between the two phytoplankton groups in + S  kRS *  a S  their half-saturation constants reported by Scavia et al.   (1988) and Padilla et al. (1996) was small relative to the difference between quartiles in Andersen (1997), which sug- where the derivatives on the right-hand side are evaluated at gests that the primary competitive difference between small M = 0. For the case in which large phytoplankton do not in- and large phytoplankton is in their maximum uptake rates. terfere with zooplankton filtering (aLTL = 0), dS*/dM =0 Although estimates of the half-saturation concentrations (eq. A2) and the second term in eq. A3 disappears. With from Scavia et al. (1988) are consistent with Andersen’s aLTL = 0, we can insert eq. A1 into eq. A3 and simplify to (1997) survey, the values used by Padilla et al. (1996) are an obtain order of magnitude larger than the upper quartile of the survey distribution. This seems surprising, because the parame- dZ*  vkvk  ter represents a composite of several species and should take (A4) =  SSLL −1 −++2 a typical value. Hence, we let k = k = 0.7 µg P·L–1, the me- dM M =0 ((kklQvkLSL )( ) LS )  S L dian of the distribution from Andersen (1997). 1 + aTS* Phytoplankton loss to sinking and other density- ×  SS b  a  independent mortality is difficult to measure. Values listed S by Scavia et al. (1988) and Padilla et al. (1996) suggest the –1 The second term in parentheses is positive if S* ≥ 0. If (k – range 0–0.05 day . Because the western basin of Lake Erie L does not stratify, phytoplankton cannot be trapped beneath kS)(lL + Q) is small relative to vLkS, the first term is positive if the thermocline and may be resuspended by turbulence. Hence, we use a value at the low end of the range and let vk –1 (A5) SL> 1 lS = lL = 0.01 day . vkLS Zooplankton This condition is eq. 8 in the text. Zooplankton require six parameters: capture rates (aS, aL) Phytoplankton density (S*+L*) must decrease with M and handling times (TS, TL) for the two phytoplankton groups, ε during the initial invasion because R* and Z* increase, and excretion rate (lZ), and assimilation efficiency ( ). Our esti-

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mates for these parameters are based on measurements of We let ε = 0.54, based on measurements of phosphorus adult D. galeata. We use data from McCauley et al. (1996), assimilation efficiency by Peters and Rigler (1973). who report carbon ingestion and loss rates. We convert their estimates to units of phosphorus. Dreissenid filtering impact We assume that the ratio of phosphorus and carbon in MacIsaac et al. (1992) estimate the whole basin filtering Daphnia tissue is approximately constant. Hence, we use the impact per unit benthic area to be bM = 13.9 m3·m–2·day–1. mass-specific maintenance rate, 0.176 day–1, reported by This value is based on their measurements of mussel density McCauley et al. (1996) for the loss parameter lZ. This value and size structure and on size-specific filtering rate measure- falls within the range of phosphorus excretion rates mea- ments from Kryger and Riisgård (1988). We divide the esti- sured directly in D. pulex by Olsen and Østgaard (1985). mate from MacIsaac et al. (1992) by the average depth of Zooplankton capture rate and handling time for edible the western basin of Lake Erie (7 m) to obtain the volume- phytoplankton can be calculated from maximum ingestion specific filtering impact, 2.0 day–1. If 15% of the benthic –1 rate (Imax) and half-saturation food concentration (FH)as surface is occupied by mussels, the local impact is 13.3 day . aS = Imax/FH and TS =1/Imax. Using the ratio P:C = 0.02 for We take our lower estimate of dreissenid filtering rate phytoplankton (Reynolds 1984), P:C = 0.038 for Daphnia from Walz (1978). Walz’s estimate of b is 0.22 times Kryger –1 (Peters and Rigler 1973), and Imax = 0.025 mg C·day and and Riisgård’s (1988) value. Hence, the local and whole- –1 –1 –1 FH = 0.98 mg C·L for D. galeata (McCauley et al. 1996), basin filtering impacts are 2.93 day and 0.44 day , respec- µ –1 we obtain aS = 0.16 L·( g P·day) and TS = 0.32 days. We tively. assume TL = TS and aL = aS to study the effects of inedible phytoplankton interference.

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